19

Let $R$ be a ring such that $x^4=x$ for every $x\in R$. Is this ring commutative?

Mamad Hardaniyan
  • 323
  • 8
    Have you tried anything so far which you could share? – Rex Butler Oct 28 '11 at 21:44
  • 6
    There is a theorem of Jacobson that you may find useful, see here http://mathoverflow.net/questions/32032/on-a-theorem-of-jacobson. Short of that I feel like the easiest path is to just mess around with expansions until you get the desired result. – Alex Youcis Oct 28 '11 at 21:47
  • 2
    The analogous problem with $3$ in place of $4$ is a classic (perhaps due to its appearance in a book by Herstein). It turns out that if there is a positive integer $n > 1$ with the property that $x^n = x$ holds for all $x$ in $R$, then $R$ must be commutative. I do not know if the $n=4$ case is easier to prove than the general case (a wild guess: maybe try to show that if it holds for $n=4$, then it holds for $n=2$). The link http://www.math.niu.edu/~rusin/known-math/99/commut_ring of some sci.math discussion of the $n=3$ problem may be of interest. – leslie townes Oct 28 '11 at 21:48
  • The n=4 case appears in Herstein as well, actually; IIRC, it appears earlier. From what I recall of the solution, it's slightly easier than the n=3 case. – Harry Altman Oct 29 '11 at 21:25

3 Answers3

30

Here's an old post of mine from Yahoo! Answers:

First, note $-x = (-x)^4 = x^4 = x$, so $x+x = 0$ for any $x$ in $R$. Then $(x^2+x)^2 = x^2 + x + x^3 + x^3 = x^2+x$. Thus $x^2+x$ is idempotent, and it is easy to see idempotent elements are central in this ring. [I give a proof of this at the end.]

Now let $x=a+b$, where $a$ and $b$ are arbitrary. From above, for any $c$ in $R$, $c(x^2 + x) = (x^2 + x)c$, and expanding this out and cancelling terms we get $c(ab + ba) = (ab + ba)c$. Setting $c=a$, we get, after cancelling again, $a^2b = ba^2$. Thus, for any $x$ in $R$, $x^2$ is central. Then of course $x = (x^2+x)-x^2$ is central.

To prove that idempotents are central, first note that if $xy=0$, then $yx = (yx)^4 = y (xy)(xy)(xy)x = 0$. So now if $z^2 = z$, then $z(y - zy) = 0$, so $(y-zy)z = 0$, or $yz = zyz$. Similarly, $(yz - y)z = 0$, so $z(yz-y) = 0$, or $zy = zyz$. Thus $yz = zy$.

  • How are we getting $c(ab+ba)=(ab+ba)c$ ? – Mohan Nov 17 '11 at 13:37
  • @user774025 Have you expanded out $c(x^2+x)=(x^2+x)c$ when $x=a+b$? –  Nov 17 '11 at 13:42
  • Yes, I'm getting $ca^2 + cb^2 + cba +cab + ca + cb= a^2c +b^2c + bac + abc + ac + bc$. I'm not understanding which terms will cancel out. – Mohan Nov 17 '11 at 13:59
  • @user774025 And of course you're using $x^2+x$ is central, right? For any $x$? :) –  Nov 17 '11 at 17:18
  • ok.I get it now. Thanks. :) – Mohan Nov 17 '11 at 17:42
  • 1
    +1 very nice, but you've gotta hate an argument like this -- opaque and full of ingenious twists. I wonder if there's anything more straightforward but still elementary. – Jonathan May 14 '13 at 05:16
  • @Jonathan: While I agree this seems somewhat opaque, this is a natural way of beginning to prove the much stronger version of this theorem (Jacobson's theorem): look at idempotents, and the differences between them. –  May 14 '13 at 19:55
  • 1
    Hi, nice argument. But when you chose to set $c=a$ then how did you deduce that for every $x$, $x^2$ is central? I think your $a$ wasn't an arbitrary element anymore. – Hoda Bibo Nov 25 '20 at 11:46
4

In my previous version of this answer, I gave an automated Prover9 proof of this result using a step-by-step approach. It turns out that this is unnecessary. Previously, I used the default Lexicographic Path Ordering, whereas the Knuth-Bendix Ordering (assign(order, kbo).)is much faster for this problem (ref.), and we can obtain a proof:

============================== PROOF =================================

% Proof 1 at 46.52 (+ 0.36) seconds.
% Length of proof is 243.
% Level of proof is 45.
% Maximum clause weight is 35.
% Given clauses 639.

1 x * y = y * x # label(non_clause) # label(goal).  [goal].
2 (x + y) + z = x + (y + z).  [assumption].
3 x + y = y + x.  [assumption].
4 x + 0 = x.  [assumption].
5 0 + x = x.  [assumption].
6 x + -x = 0.  [assumption].
8 x * (y + z) = (x * y) + (x * z).  [assumption].
9 (x * y) + (x * z) = x * (y + z).  [copy(8),flip(a)].
10 (x + y) * z = (x * z) + (y * z).  [assumption].
11 (x * y) + (z * y) = (x + z) * y.  [copy(10),flip(a)].
12 (x * y) * z = x * (y * z).  [assumption].
13 x * (x * (x * x)) = x.  [assumption].
14 c2 * c1 != c1 * c2.  [deny(1)].
15 x + (y + z) = y + (x + z).  [para(3(a,1),2(a,1,1)),rewrite([2(2)])].
16 x + (-x + y) = y.  [para(6(a,1),2(a,1,1)),rewrite([5(2)]),flip(a)].
18 -x + (y + x) = y.  [para(6(a,1),2(a,2,2)),rewrite([3(3),4(5)])].
20 (x * y) + ((x * z) + u) = (x * (y + z)) + u.  [para(9(a,1),2(a,1,1)),flip(a)].
21 (x * y) + ((z * y) + u) = ((x + z) * y) + u.  [para(11(a,1),2(a,1,1)),flip(a)].
22 (x + x) * y = x * (y + y).  [para(11(a,1),9(a,1))].
23 (x * (y * z)) + (u * z) = ((x * y) + u) * z.  [para(12(a,1),11(a,1,1))].
24 (x * y) + (z * (u * y)) = (x + (z * u)) * y.  [para(12(a,1),11(a,1,2))].
25 x * ((x * (x * x)) + y) = x + (x * y).  [para(13(a,1),9(a,1,1)),flip(a)].
26 x * (y + (x * (x * x))) = x + (x * y).  [para(13(a,1),9(a,1,2)),rewrite([3(2)]),flip(a)].
27 (x + y) * (x * (x * x)) = x + (y * (x * (x * x))).  [para(13(a,1),11(a,1,1)),flip(a)].
29 x * (x * (x * (x * y))) = x * y.  [para(13(a,1),12(a,1,1)),rewrite([12(4),12(3)]),flip(a)].
30 x * (y * (x * (y * (x * (y * (x * y)))))) = x * y.  [para(13(a,1),12(a,1)),rewrite([12(5),12(6)]),flip(a)].
31 x + (y + -x) = y.  [para(6(a,1),15(a,1,2)),rewrite([4(2)]),flip(a)].
32 (x * y) + (z + (x * u)) = z + (x * (y + u)).  [para(9(a,1),15(a,1,2)),flip(a)].
36 --x = x.  [para(6(a,1),16(a,1,2)),rewrite([4(2)]),flip(a)].
39 -(x * y) + (x * (z + y)) = x * z.  [para(9(a,1),18(a,1,2))].
43 x + -(x + y) = -y.  [para(18(a,1),18(a,1,2)),rewrite([3(3)])].
49 x + (y + (z + -x)) = y + z.  [para(2(a,1),31(a,1,2))].
58 (x * y) + -(x * (y + z)) = -(x * z).  [para(9(a,1),43(a,1,2,1))].
66 (x * (y + z)) + (u * z) = (x * y) + ((x + u) * z).  [para(11(a,1),20(a,1,2)),flip(a)].
73 0 * (x + x) = 0 * x.  [para(4(a,1),22(a,1,1)),flip(a)].
74 (x + x) * 0 = x * 0.  [para(4(a,1),22(a,2,2))].
98 0 * (x + (x + y)) = 0 * (x + y).  [para(73(a,1),9(a,1,1)),rewrite([9(5),2(6)]),flip(a)].
105 (x + (x + y)) * 0 = (x + y) * 0.  [para(74(a,1),11(a,1,1)),rewrite([11(5),2(5)]),flip(a)].
114 ((x + y) * z) + (y * u) = (x * z) + (y * (z + u)).  [para(9(a,1),21(a,1,2)),flip(a)].
116 (x * y) + ((z * (u * y)) + w) = ((x + (z * u)) * y) + w.  [para(12(a,1),21(a,1,2,1))].
119 -(x * y) + ((x + z) * y) = z * y.  [para(21(a,1),31(a,1)),rewrite([3(5)])].
143 x * ((x * (x * (x * y))) + z) = x * (y + z).  [para(29(a,1),9(a,1,1)),rewrite([9(3)]),flip(a)].
144 x * (y + (x * (x * (x * z)))) = x * (y + z).  [para(29(a,1),9(a,1,2)),rewrite([9(3)]),flip(a)].
146 x * (y * (x * (y * (x * (y * (x * (y * z))))))) = x * (y * z).  [para(29(a,1),12(a,1)),rewrite([12(2),12(6),12(7),12(8)]),flip(a)].
154 (x + (x * y)) * z = x * (z + (y * z)).  [para(23(a,1),9(a,1)),rewrite([3(2),3(5)])].
155 (x * (y * (z * u))) + (w * u) = ((x * (y * z)) + w) * u.  [para(12(a,1),23(a,1,1,2))].
156 (x * (y * z)) + (u * (w * z)) = ((x * y) + (u * w)) * z.  [para(12(a,1),23(a,1,2))].
157 ((x * y) + z) * (y * (y * y)) = (x * y) + (z * (y * (y * y))).  [para(13(a,1),23(a,1,1,2)),flip(a)].
158 ((x * x) + y) * (x * x) = x + (y * (x * x)).  [para(13(a,1),23(a,1,1)),flip(a)].
159 (x * (y * z)) + (u + (w * z)) = u + (((x * y) + w) * z).  [para(23(a,1),15(a,1,2)),flip(a)].
173 (x * (y * z)) + (u * (y * (y * (y * z)))) = ((x * y) + u) * (y * (y * (y * z))).  [para(29(a,1),23(a,1,1,2))].
174 ((x * x) + y) * (x * (x * z)) = (x * z) + (y * (x * (x * z))).  [para(29(a,1),23(a,1,1)),flip(a)].
178 0 * 0 = 0 * x.  [para(6(a,1),98(a,1,2,2)),rewrite([4(3),6(5)]),flip(a)].
191 0 * 0 = 0.  [para(178(a,1),13(a,1,2,2)),rewrite([178(5,R),178(5,R)])].
193 0 * x = 0.  [para(178(a,2),22(a,2)),rewrite([4(3),191(5)])].
198 x * 0 = 0.  [para(6(a,1),105(a,1,1,2)),rewrite([4(2),6(4),193(5)])].
199 (x * y) + (z * (u * (w * y))) = (x + (z * (u * w))) * y.  [para(12(a,1),24(a,1,2,2))].
200 (x + (y * z)) * (z * (z * z)) = ((x * (z * z)) + y) * z.  [para(13(a,1),24(a,1,2,2)),rewrite([155(5)]),flip(a)].
201 (x + (y * y)) * (y * y) = y + (x * (y * y)).  [para(13(a,1),24(a,1,2)),rewrite([3(3)]),flip(a)].
202 (x * y) + (z + (u * (w * y))) = z + ((x + (u * w)) * y).  [para(24(a,1),15(a,1,2)),flip(a)].
215 (x + (y * y)) * (y * (y * z)) = (y + (x * (y * y))) * z.  [para(29(a,1),24(a,1,2)),rewrite([3(5),199(5)]),flip(a)].
216 ((x * x) + y) * (x * (x * z)) = (x + (y * (x * x))) * z.  [back_rewrite(174),rewrite([199(10)])].
217 ((x * y) + z) * (y * (y * (y * u))) = (x + (z * (y * y))) * (y * u).  [back_rewrite(173),rewrite([199(7)]),flip(a)].
218 ((x * y) + z) * (y * (y * y)) = (x + (z * (y * y))) * y.  [back_rewrite(157),rewrite([199(10)])].
225 -(x * y) = x * -y.  [para(18(a,1),39(a,1,2,2)),rewrite([3(5),58(5)])].
242 (x * -y) + ((x + z) * y) = z * y.  [back_rewrite(119),rewrite([225(2)])].
247 x * (x * (x * -x)) = -x.  [para(13(a,1),225(a,1,1)),rewrite([225(4),225(3)]),flip(a)].
256 x * (y * (x * (y * (x * (y * (x * -y)))))) = x * -y.  [para(247(a,1),12(a,1)),rewrite([225(2),225(6),12(7),12(8)]),flip(a)].
257 -x * (-x * (-x * x)) = x.  [para(36(a,1),247(a,1,2,2,2)),rewrite([36(8)])].
268 x * (x * ((x * x) + y)) = x + (x * (x * y)).  [para(9(a,1),25(a,1,2))].
272 x * (((x * (x * x)) + y) * z) = (x + (x * y)) * z.  [para(25(a,1),12(a,1,1)),flip(a)].
305 x * (-y * (x * (-y * (x * (-y * (x * y)))))) = x * y.  [para(225(a,1),257(a,1,1)),rewrite([225(4),225(6),12(8),12(9),12(10)])].
326 -x * y = x * -y.  [para(6(a,1),242(a,1,2,1)),rewrite([193(4),3(4),5(4)]),flip(a)].
348 x * -y = x * y.  [back_rewrite(305),rewrite([326(5),225(4),326(7),225(6),225(5),225(4),36(3),326(7),225(6),225(5),225(4),225(3),225(2),256(8)])].
369 -x = x.  [back_rewrite(247),rewrite([348(2),13(3)]),flip(a)].
399 x + (y + (z + x)) = y + z.  [back_rewrite(49),rewrite([369(1)])].
401 x + (y + x) = y.  [back_rewrite(31),rewrite([369(1)])].
402 x + (x + y) = y.  [back_rewrite(16),rewrite([369(1)])].
403 x + x = 0.  [back_rewrite(6),rewrite([369(1)])].
405 x * (x * (y + (x * x))) = x + (x * (x * y)).  [para(9(a,1),26(a,1,2))].
408 x * ((y + (x * (x * x))) * z) = (x + (x * y)) * z.  [para(26(a,1),12(a,1,1)),flip(a)].
412 x + ((x * y) + (z * (u * (y + (x * (x * x)))))) = ((z * u) + x) * (y + (x * (x * x))).  [para(26(a,1),23(a,1,2)),rewrite([3(8),2(8)])].
414 x * ((y + (x * x)) * x) = x + (x * (y * x)).  [para(24(a,1),26(a,1,2))].
416 x + ((y * (x * (x * x))) + ((x + y) * z)) = (x + y) * ((x * (x * x)) + z).  [para(27(a,1),9(a,1,1)),rewrite([2(7)])].
422 (x + (y * (x * (x * x)))) * z = (x + y) * (x * (x * (x * z))).  [para(27(a,1),12(a,1,1)),rewrite([12(9),12(8)])].
444 (x + (x * y)) * (y * z) = x * (y * (z + (y * z))).  [para(9(a,1),154(a,2,2))].
449 (x + (x * y)) * (z * u) = x * ((z + (y * z)) * u).  [para(154(a,1),12(a,1,1)),rewrite([12(4)]),flip(a)].
451 (x + (x * (y * z))) * u = x * (u + (y * (z * u))).  [para(12(a,1),154(a,2,2,2))].
564 (x * y) + ((z + x) * u) = (z * u) + (x * (y + u)).  [para(11(a,1),32(a,1,2))].
566 ((x + y) * z) + (x * u) = (y * z) + (x * (z + u)).  [para(32(a,1),21(a,1)),flip(a)].
602 x * (x * ((x * (x * y)) + z)) = x * (y + (x * z)).  [para(9(a,1),143(a,1,2))].
636 x * (x * (y + (x * (x * z)))) = x * ((x * y) + z).  [para(9(a,1),144(a,1,2))].
805 ((x + y) * z) + (x * u) = (x * (u + z)) + (y * z).  [para(66(a,2),3(a,1)),flip(a)].
839 (x * (y + z)) + ((u + x) * z) = (x * y) + (u * z).  [para(401(a,1),66(a,2,2,1))].
931 x * (y + ((z + (x * x)) * x)) = x + (x * (y + (z * x))).  [para(414(a,1),9(a,1,2)),rewrite([15(5),9(4)]),flip(a)].
933 x * ((y + (x * x)) * (x * z)) = (x + (x * (y * x))) * z.  [para(414(a,1),12(a,1,1)),rewrite([12(8)]),flip(a)].
940 (x + (y * (z + (y * y)))) * y = y + ((x + (y * z)) * y).  [para(414(a,1),24(a,1,2)),rewrite([15(5),24(4)]),flip(a)].
997 ((x + y) * z) + (y * (u + z)) = (x * z) + (y * u).  [para(401(a,1),114(a,2,2,2))].
998 (x * y) + ((z + x) * u) = (z * y) + ((z + x) * (y + u)).  [para(402(a,1),114(a,1,1,1))].
1233 (x + (x * (x * x))) * x = x + (x * x).  [para(272(a,1),414(a,1)),rewrite([12(7),12(6),13(7)])].
1248 (x + (x * (x * x))) * (x * y) = (x + (x * x)) * y.  [para(1233(a,1),12(a,1,1)),flip(a)].
1256 (x + (x * (x * x))) * (y + (x * y)) = x * ((x + (x * x)) * y).  [para(1233(a,1),154(a,1,1,2)),rewrite([3(6),15(6),2(5),9(4),402(5),12(4)]),flip(a)].
1876 (x + (x * x)) * (y + (x * y)) = (x + (x * (x * x))) * y.  [para(1233(a,1),444(a,1,1,2)),rewrite([3(6),15(6),2(5),9(4),402(5),12(5),933(5),1248(11)]),flip(a)].
2110 (x + (x * (x * x))) * ((y + (x * y)) * z) = x * ((x + (x * x)) * (y * z)).  [para(1233(a,1),449(a,1,1,2)),rewrite([3(6),15(6),2(5),9(4),402(5),12(5)]),flip(a)].
7294 ((x * (x * x)) + y) * x = x + (y * x).  [para(13(a,1),155(a,1,1)),flip(a)].
7299 ((x * (x * x)) + y) * (x * z) = (x + (y * x)) * z.  [para(29(a,1),155(a,1,1)),rewrite([24(4)]),flip(a)].
7543 (x + (x * (y * (y * y)))) * y = 0.  [para(7294(a,1),408(a,1,2)),rewrite([12(3),12(2),144(5),403(1),198(2)]),flip(a)].
7634 (x + (x * (y * (y * y)))) * (y * z) = 0.  [para(7543(a,1),12(a,1,1)),rewrite([193(2)]),flip(a)].
7788 x * (y + (y * (x * (x * x)))) = 0.  [para(7634(a,1),30(a,1,2,2,2,2,2)),rewrite([198(10),198(10),198(6),198(6),198(2)]),flip(a)].
7809 x * ((y + (y * (x * (x * x)))) * z) = 0.  [para(7634(a,1),146(a,1,2,2,2,2,2)),rewrite([198(10),198(10),198(6),198(6),198(2)]),flip(a)].
7963 x * (y + ((y * (x * (x * x))) + z)) = x * z.  [para(7788(a,1),9(a,1,1)),rewrite([5(3),2(6)]),flip(a)].
7968 x * (y * (z + (z * (x * (x * x))))) = 0.  [para(12(a,1),7788(a,1,2,2)),rewrite([9(6)])].
8032 x * (y * (z + (x * (x * (x * z))))) = 0.  [para(154(a,1),7809(a,1,2)),rewrite([12(3),12(2)])].
8122 (x + (x * (y * (y * y)))) * (z * y) = 0.  [para(7809(a,1),7968(a,1,2,2,2)),rewrite([4(6)])].
8399 x * ((y + (z * (z * (z * y)))) * z) = 0.  [para(8122(a,1),449(a,1)),rewrite([12(4),12(3)]),flip(a)].
8445 (x + (y * (y * (y * x)))) * y = 0.  [para(8399(a,1),13(a,1,2,2)),rewrite([198(12),198(7)]),flip(a)].
8493 (x + (y * (y * (y * x)))) * (z * y) = 0.  [para(8399(a,1),8032(a,1,2,2,2,2)),rewrite([198(10),4(6)])].
8495 (x + ((y * (y * (y * x))) + z)) * y = z * y.  [para(8445(a,1),11(a,1,1)),rewrite([5(3),2(6)]),flip(a)].
8498 (x + (y * (y * (y * x)))) * (y * z) = 0.  [para(8445(a,1),12(a,1,1)),rewrite([193(2)]),flip(a)].
8583 (x + (y * (y * (y * x)))) * (z * (u * y)) = 0.  [para(12(a,1),8493(a,1,2))].
9132 (x + (y + (z * (z * (z * (x + y)))))) * (z * u) = 0.  [para(2(a,1),8498(a,1,1))].
9199 (x * (x * (x * (y * z)))) + (y * (x * u)) = x * (x * (x * (y * (z + (x * u))))).  [para(8498(a,1),839(a,1,2)),rewrite([12(6),12(5),12(4),3(8),5(8),12(10),12(9),12(8)]),flip(a)].
9862 x * (y + (y * ((x * (x * x)) + z))) = x * (y * z).  [para(9(a,1),7963(a,1,2,2))].
9875 x * (y * (x * (x * x))) = x * y.  [para(403(a,1),7963(a,1,2,2)),rewrite([4(2)]),flip(a)].
9876 x * (y + (z * (x * (x * x)))) = x * (z + y).  [para(401(a,1),7963(a,1,2,2)),flip(a)].
9877 x * ((y * (x * (x * x))) + z) = x * (y + z).  [para(402(a,1),7963(a,1,2,2)),flip(a)].
9878 x * (y + (z + (u * (x * (x * x))))) = x * (u + (y + z)).  [para(399(a,1),7963(a,1,2,2)),flip(a)].
9946 x * (y * (z + (x * (x * x)))) = x * (y + (y * z)).  [para(564(a,2),7963(a,1,2,2)),rewrite([9878(8),15(4),2(3),402(4)]),flip(a)].
10001 x * (y * (x * (x * (x * z)))) = x * (y * z).  [para(9875(a,1),12(a,1,1)),rewrite([12(2),12(6),12(5),12(4)]),flip(a)].
10003 x * (y * (z * (x * (x * x)))) = x * (y * z).  [para(12(a,1),9875(a,1,2))].
10009 (x + (y * z)) * (y * (y * y)) = (x * (y * (y * y))) + (y * z).  [para(9875(a,1),24(a,1,2)),flip(a)].
10077 x * (y * (x * (y * (x * (y * x))))) = x * (y * (y * y)).  [para(9875(a,1),146(a,1,2,2,2,2,2))].
10274 (x + (y * ((z + (u * u)) * u))) * u = (x * u) + (y * (u + (z * (u * u)))).  [para(201(a,1),199(a,1,2,2)),flip(a)].
10415 (x + (y * (z * u))) * (z * (z * z)) = (x * (z * (z * z))) + (y * (z * u)).  [para(9875(a,1),199(a,1,2,2)),flip(a)].
10416 (x + (y * (z * y))) * (y * y) = (x * (y * y)) + (y * z).  [para(9875(a,1),199(a,1,2)),flip(a)].
10973 (x * (y * (y * y))) + (y * (y * (y * x))) = 0.  [para(8445(a,1),200(a,2)),rewrite([12(6),12(5),12(4),12(3),12(2),9875(4),10009(7)])].
10988 x * (((y * (x * x)) + z) * x) = x * (y + (z * x)).  [para(200(a,1),9875(a,1,2))].
11257 x * (x * (x * (y * x))) = y * x.  [para(403(a,1),8495(a,1,1,2)),rewrite([4(2),12(5),12(4),12(3)]),flip(a)].
11259 ((x * (x * (x * y))) + z) * x = (y + z) * x.  [para(402(a,1),8495(a,1,1,2)),flip(a)].
11383 (x + (((y * (y * y)) + z) * x)) * y = z * (x * y).  [para(155(a,1),8495(a,1,1,2)),rewrite([12(8)])].
11436 x * (y + (x * (x * (z * x)))) = (x * y) + (z * x).  [para(11257(a,1),9(a,1,2)),flip(a)].
11439 x * (x * (x * (y * (x * z)))) = y * (x * z).  [para(11257(a,1),12(a,1,1)),rewrite([12(2),12(6),12(5),12(4)]),flip(a)].
11441 x * (x * (x * (y * (z * x)))) = y * (z * x).  [para(12(a,1),11257(a,1,2,2,2)),rewrite([12(7)])].
11446 (x + (y * z)) * (z * (z * (u * z))) = ((x * (z * z)) + y) * (u * z).  [para(11257(a,1),24(a,1,2,2)),rewrite([155(7)]),flip(a)].
11550 (x + (y * (y * (z * (z * (z * (y * x))))))) * (z * y) = 0.  [para(11257(a,1),8493(a,1,2)),rewrite([12(10),12(9),12(8),12(11),12(10),12(9),11439(8),12(9),12(8),12(7),10001(8)])].
12710 ((x + y) * (z * (x * (x * x)))) + (x * u) = (x * (z + u)) + (y * (z * (x * (x * x)))).  [para(9876(a,1),805(a,2,1))].
12715 (x * (y + z)) + ((u + x) * (y * (x * (x * x)))) = (x * z) + (u * (y * (x * (x * x)))).  [para(9876(a,1),839(a,1,1))].
12726 ((x + y) * (z * (y * (y * y)))) + (y * (z + u)) = (x * (z * (y * (y * y)))) + (y * u).  [para(9876(a,1),997(a,1,2))].
12837 ((x + y) * (z * (x * (x * x)))) + (x * u) = (y * (z * (x * (x * x)))) + (x * (z + u)).  [para(9877(a,1),566(a,2,2))].
12904 x * (y * (z * (x * (x * (x * u))))) = x * (y * (z * u)).  [para(12(a,1),10001(a,1,2)),rewrite([12(8)])].
12906 (x * (y * (z * u))) + (w * (z * (y * (y * (y * u))))) = ((x * y) + w) * (z * (y * (y * (y * u)))).  [para(10001(a,1),23(a,1,1,2))].
12912 (x + y) * (z * ((x + y) * ((x + y) * (x + (y * (x * (x * x))))))) = (x + y) * (z * (x * (x * x))).  [para(27(a,1),10001(a,1,2,2,2,2))].
12923 x * (y * (z + (x * (x * (x * u))))) = x * (y * (z + u)).  [para(144(a,1),10001(a,1,2,2,2,2)),rewrite([10001(6)]),flip(a)].
12934 x * (y * (x * (x * ((x + (x * z)) * u)))) = x * (y * (((x * (x * x)) + z) * u)).  [para(272(a,1),10001(a,1,2,2,2,2))].
12980 x * (y * (x * (x * (z + (x * u))))) = x * (y * ((x * (x * z)) + u)).  [para(602(a,1),10001(a,1,2,2,2))].
12987 x * (y * (x * (y * (x * (y * (x * z)))))) = x * (y * (y * (y * z))).  [para(10001(a,1),146(a,1,2,2,2,2,2))].
13027 x * ((y + (x * (z * (x * (z * (x * (z * y))))))) * z) = 0.  [para(8583(a,1),10001(a,1,2)),rewrite([198(2),12(5),12(6),12(7)]),flip(a)].
13083 x * (y * (z * (u * (x * (x * x))))) = x * (y * (z * u)).  [para(12(a,1),10003(a,1,2,2))].
13089 (x * (y * (z * (z * z)))) + (z * (u * y)) = (x + (z * u)) * (y * (z * (z * z))).  [para(10003(a,1),24(a,1,2))].
13439 x * (y * (y * y)) = y * (y * (y * x)).  [para(10973(a,1),401(a,1,2)),rewrite([3(5),5(5)]),flip(a)].
13460 x * (y * (y * (y * z))) = y * (y * (y * (x * z))).  [para(10973(a,1),66(a,2,2,1)),rewrite([12(5),12(4),12(3),12(9),12(8),12(7),3(10),9199(10),12923(7),401(2),12(8),12(7),12(6),193(10),3(10),5(10)]),flip(a)].
13517 (x * (y * (y * y))) + (y * z) = y * ((y * (y * x)) + z).  [para(13439(a,2),9(a,1,1))].
13548 (x + y) * ((x + y) * (x + (y * (x * (x * x))))) = x * (x * (x * ((x + y) * ((x + y) * (x + y))))).  [para(27(a,1),13439(a,2,2,2)),rewrite([12(8),12(7)]),flip(a)].
13574 (x + (y * (x * x))) * x = x * (x + (x * (x * y))).  [para(268(a,1),13439(a,2,2)),rewrite([216(5)])].
13623 (x + y) * (x * (x * (x * z))) = x * (x * (x * ((y + x) * z))).  [para(13439(a,2),408(a,2,1,2)),rewrite([9(5),9(3),12(4),12(3),422(10)]),flip(a)].
13848 x * (y * (z * (z * (z * (x * (x * x)))))) = x * (z * (z * (z * y))).  [para(13439(a,1),10003(a,2,2)),rewrite([12(5),12(4)])].
13863 (x + (y * z)) * (y * (y * y)) = y * ((y * (y * x)) + z).  [back_rewrite(10009),rewrite([13517(10)])].
13885 (x + y) * (z * (x * (x * x))) = x * (x * (x * ((y + x) * z))).  [back_rewrite(12912),rewrite([13548(9),13848(11),13623(5)]),flip(a)].
14123 (x * (y * (z * (z * z)))) + (z * (y + u)) = z * ((z * (z * ((x + z) * y))) + u).  [back_rewrite(12837),rewrite([13885(5),9(7)]),flip(a)].
14124 (x * (y + z)) + (u * (y * (x * (x * x)))) = x * ((x * (x * ((u + x) * y))) + z).  [back_rewrite(12710),rewrite([13885(5),9(7)]),flip(a)].
14224 (x * (y * (z * (z * z)))) + (z * u) = z * ((z * (z * (x * y))) + u).  [back_rewrite(12726),rewrite([14123(8),3(2),401(2)]),flip(a)].
14226 (x * y) + (z * (u * (x * (x * x)))) = x * ((x * (x * (z * u))) + y).  [back_rewrite(12715),rewrite([14124(8),3(2),401(2)]),flip(a)].
14262 (x + (y * z)) * (u * (y * (y * y))) = y * (((y * (y * x)) + z) * u).  [back_rewrite(13089),rewrite([14224(7),155(5)]),flip(a)].
14429 x * (((x * (x * y)) + z) * x) = (y + (x * z)) * x.  [para(9(a,1),11259(a,1,1)),rewrite([12(5)])].
14684 (x + (y * (x * x))) * (z + (x * (x * u))) = (y + (x * x)) * (x * ((x * z) + u)).  [para(636(a,1),215(a,1,2)),flip(a)].
14786 (x + (y * (x * x))) * (x * z) = x * ((x + (x * (x * y))) * z).  [para(13439(a,2),215(a,1,2)),rewrite([14262(6),3(3)]),flip(a)].
14838 x * (x * ((x + (x * y)) * (x * z))) = (x + (y * x)) * z.  [para(272(a,1),11439(a,1,2,2)),rewrite([7299(11)])].
15052 x * (y * (x * (y * (x * (y * y))))) = x * (x * (x * y)).  [para(13439(a,2),11441(a,1,2,2,2)),rewrite([12(7),10003(7),12(5),12(6)])].
16018 ((x * y) + z) * (u * (y * (y * (y * w)))) = (x + (z * (u * (y * (u * (y * u)))))) * (y * (u * w)).  [para(23(a,1),217(a,1,1)),rewrite([12(7),12(8),12(9),12(10),12987(9),12(11),12(12),12(15)])].
16020 (x + (((y * z) + u) * z)) * (z * (z * w)) = (y + ((x + (u * z)) * z)) * (z * w).  [para(217(a,1),24(a,1,2)),rewrite([23(9),202(5)]),flip(a)].
16999 (x + (((y * z) + u) * z)) * (z * z) = (y + ((x + (u * z)) * z)) * z.  [para(218(a,1),24(a,1,2)),rewrite([23(7),202(5)]),flip(a)].
17991 (x + (y * x)) * (x * x) = x * (x * (x + (x * y))).  [para(7299(a,1),13439(a,1)),rewrite([25(8)])].
20339 x * (y * (z + (z * ((x * (x * x)) + u)))) = x * (y * (z * u)).  [para(12(a,1),9862(a,1,2,2)),rewrite([9(7),12(9)])].
20347 x * (y * ((x * (x * x)) + y)) = (x + (x * y)) * y.  [para(9862(a,1),154(a,2)),rewrite([25(4),402(3),12(5),272(10)])].
20738 x * (y * (y + (x * (x * x)))) = (x + (x * y)) * y.  [para(9946(a,2),154(a,2)),flip(a)].
21673 x * ((y + (((z * x) + u) * x)) * x) = x * (z + ((y + (u * x)) * x)).  [para(159(a,1),10988(a,1,2,1))].
21795 (x + (y + (((z * (z * z)) + u) * (x + y)))) * z = u * ((x + y) * z).  [para(2(a,1),11383(a,1,1))].
22013 (x + (y * x)) * y = (y + (y * (y * y))) * (x * y).  [para(11436(a,1),451(a,2)),rewrite([3(9),24(9)]),flip(a)].
23566 ((x * y) + z) * (u * (y * (y * (y * w)))) = (x + (z * (y * y))) * (y * (u * w)).  [para(13460(a,2),199(a,1,2,2)),rewrite([12906(9)])].
23724 (x + (y * (z * (u * (z * (u * z)))))) * (u * (z * w)) = (x + (y * (u * u))) * (u * (z * w)).  [back_rewrite(16018),rewrite([23566(7)]),flip(a)].
23995 (x * (y * z)) + (z * (z + (z * (z * u)))) = ((x * y) + (z + (u * (z * z)))) * z.  [para(13574(a,1),23(a,1,2))].
24414 x * (((x * (x * y)) + z) * (x * u)) = (y + (x * z)) * (x * u).  [para(14429(a,1),12(a,1,1)),rewrite([12(4),12(9)]),flip(a)].
24599 (x + (y * y)) * (y * (y + (y * y))) = y + ((y + (x * (y + (y * y)))) * y).  [para(13574(a,1),416(a,1,2,2)),rewrite([12(5),12(4),13(3),23995(7),15(5),9(4),3(12),14684(13),3(10)]),flip(a)].
24923 x * (y * (((x * (x * x)) + y) * z)) = (x + (x * y)) * (y * z).  [para(20347(a,1),12(a,1,1)),rewrite([12(4),12(9)]),flip(a)].
25111 (x + (x * y)) * (y + (x * (x * x))) = x + (x * (y * y)).  [para(20738(a,1),412(a,1,2,2)),rewrite([11(5),402(3),12(2),3(5)]),flip(a)].
25390 (x + (x * x)) * (y * x) = y * (x * (x + (x * x))).  [para(22013(a,2),13439(a,2,2,2)),rewrite([14684(11),3(8),24599(10),940(9),402(9),2110(8),17991(5),405(5),26(5),12(4),2110(14),14786(13),2110(12),933(10),408(10)]),flip(a)].
25416 (x + (x * (x * x))) * (y + (y * (x * x))) = (y + (x * y)) * x.  [para(22013(a,2),9862(a,2)),rewrite([14684(13),3(10),24599(12),940(11),402(11),2110(10),17991(7),405(7),26(7),3(6),402(6)])].
25444 (x + (x * x)) * (y * x) = y * ((x + (x * x)) * x).  [para(22013(a,2),13460(a,2,2,2)),rewrite([13574(10),26(10),1256(9),414(7),25416(7),2110(14),14786(13),2110(12),933(10),408(10)]),flip(a)].
26061 (x + (x * y)) * (y * y) = y * ((x + (y * x)) * y).  [para(9875(a,1),25390(a,1,2)),rewrite([12(10),12(9),12(8),9875(7),9(8),9(6),9(4),12(7),12(6),12(5),14838(6),12(17),12(16),12(15),9875(14),9(15),9(13),9(11),12(13),12(12),12(11),12980(10),20347(9),12934(11),24923(10)]),flip(a)].
26098 x * ((x + (x * x)) * ((x + (x * (x * x))) * y)) = y * (x + (x * x)).  [para(25390(a,2),13439(a,1,2,2)),rewrite([12(9),414(9),25111(9),12(7),25111(6),26(4),12(13),12(14),933(14),2110(13),933(11),12(11)]),flip(a)].
26176 (x + (x * x)) * (y + (y * (x * x))) = y * (x * (x + (x * x))).  [para(25390(a,1),9862(a,2)),rewrite([1876(9),13574(8),26(8),1876(7),13574(6),26(6),3(5),402(5)])].
26449 (x + (x * x)) * (y * (x + (x * x))) = y * (x + (x * x)).  [para(25444(a,2),13439(a,1,2,2)),rewrite([12(9),26061(9),414(9),26176(9),405(7),12(7),26(6),1876(5),13574(4),26(4),12(13),12(14),933(13),12(14),26098(13)]),flip(a)].
26578 (x + (x * x)) * y = y * (x + (x * x)).  [para(25444(a,2),20347(a,2)),rewrite([12(10),931(9),20339(10),27(5),12(4),13(3),10274(11),9(11),402(9),14262(9),3(6),408(8)]),flip(a)].
26672 (x * y) + (y * (z + (z * z))) = (x + (z + (z * z))) * y.  [para(26578(a,1),11(a,1,2))].
26673 x * (y + (y * y)) = y * (x + (y * x)).  [para(26578(a,1),11(a,2)),rewrite([12(3),9(4)]),flip(a)].
26725 (x + (x * x)) * (y + (z * (x + (x * x)))) = (x + (x * x)) * (y + z).  [para(26578(a,1),144(a,1,2,2,2,2)),rewrite([26449(10),26449(8)])].
26859 (x + (x * x)) * (y + ((x + (x * x)) * z)) = (x + (x * x)) * (y + z).  [para(26578(a,1),602(a,2,2,2)),rewrite([602(13),26725(14)])].
27111 (x + (x * y)) * y = y * (x + (y * x)).  [para(26578(a,2),9862(a,1)),rewrite([2(11),21795(12),7294(4),272(8)]),flip(a)].
27820 (x * (y + (x * y))) + (z * x) = (y + ((y * x) + z)) * x.  [para(27111(a,1),11(a,1,1)),rewrite([2(8)])].
27822 (x + (x * y)) * (y * z) = y * ((x + (y * x)) * z).  [para(27111(a,1),12(a,1,1)),rewrite([12(4)]),flip(a)].
28041 (x + (y * (x + (y * x)))) * (y + (y * y)) = 0.  [para(26673(a,1),27111(a,1,1,2)),rewrite([26859(14),403(10),198(11)])].
28298 (x + (x * x)) * (y + (x * (y + (x * y)))) = 0.  [para(28041(a,1),26578(a,2))].
28363 (x + (y * (x + (y * x)))) * ((y + (y * y)) * z) = 0.  [para(28298(a,1),9132(a,1,1,2,2,2,2)),rewrite([198(9),198(7),3(5),5(5)])].
31439 (x + (y * (x + (y * x)))) * (y + (y * (y * y))) = 0.  [para(158(a,1),28363(a,1,2)),rewrite([12(3),12(5),636(5),3(2),12(7),13(7)])].
37695 x * (y + (((y * x) + (x * y)) * x)) = y * x.  [para(31439(a,1),998(a,2,2)),rewrite([12(4),13863(11),15(9),3(8),9(8),9(10),15(8),3(7),27820(7),15(4),402(5),3(9),5(9)])].
37721 x * (y + ((y + (x * (y * x))) * x)) = y * (x * x).  [para(11(a,1),37695(a,1,2)),rewrite([12(2),156(5),21673(7),12(2),12(8)])].
37722 x * ((y + (((y * x) + (x * y)) * x)) * z) = y * (x * z).  [para(37695(a,1),12(a,1,1)),rewrite([12(2)]),flip(a)].
37942 x * ((y + ((y + (x * (y * x))) * x)) * z) = y * (x * (x * z)).  [para(37721(a,1),12(a,1,1)),rewrite([12(3),12(2)]),flip(a)].
43797 (x + ((y + (z * (u * z))) * z)) * z = ((x + (y * z)) * z) + (z * u).  [para(10416(a,1),24(a,1,2)),rewrite([116(6)]),flip(a)].
43804 (x + ((x + (y * z)) * y)) * (y * y) = y * (((x + (y * x)) * y) + z).  [para(27822(a,1),10416(a,2,1)),rewrite([2(5),24(4),9(12)])].
44721 ((x * x) + (y * (x * (y * (x * y))))) * (y * x) = 0.  [para(10077(a,1),11550(a,1,1,2,2,2,2,2)),rewrite([13083(10),14226(9),12(10),24414(10)])].
44779 x * (((x * x) + (y * (x * (y * (x * y))))) * y) = 0.  [para(10077(a,1),44721(a,1,2)),rewrite([12(11),12(10),12(9),12(8),12(7),10077(6),9875(4),12(16),12(15),12(14),12(13),12(12),15052(16),12(15),12(14),12(13),12(12),12(11),12904(11),9875(9),23724(15),3(8),14262(12),14224(9),12(9),14429(9)])].
46495 x * (y * ((x * x) + (y * (x * (y * (x * y)))))) = 0.  [para(44779(a,1),13027(a,1,2,1,2,2,2,2,2)),rewrite([198(14),198(8),198(8),198(2),4(2)])].
46810 x * ((y * y) + (x * (y * (x * (y * x))))) = 0.  [para(46495(a,1),13439(a,2,2,2)),rewrite([12(10),10415(9),12(4),13(3),198(9),198(9)])].
46856 x * ((x * (x * (y * y))) + (y * (x * (y * x)))) = 0.  [para(46810(a,1),13439(a,2,2,2)),rewrite([13863(9),198(10),198(10)])].
47125 (x * (x * (y * y))) + (y * (x * (y * x))) = 0.  [para(46856(a,1),13439(a,2,2,2)),rewrite([10415(10),12(6),12(5),12(4),10003(5),198(9),198(9)])].
47157 x * (y * (x * y)) = y * (y * (x * x)).  [para(47125(a,1),401(a,1,2)),rewrite([3(5),5(5)])].
47247 x * (x * (y * (x * (y * x)))) = x * (y * y).  [para(47157(a,2),29(a,1,2,2))].
47365 x * (x * (y * (y * (x * x)))) = y * (x * y).  [para(47157(a,1),13439(a,2,2,2)),rewrite([12(5),12(4),9875(4)]),flip(a)].
47370 x * (x * (y * (x * (x * (y * x))))) = x * (y * (x * y)).  [para(47157(a,2),11439(a,1,2)),rewrite([12(3),12(5),12(9)])].
47501 (x + ((x + (y * (x * y))) * y)) * (y * x) = x * (y * (x * y)).  [para(37695(a,1),47157(a,1,2,2)),rewrite([37722(8),16999(15),12(10),43797(14),27111(11),3(13),9(13),402(11),16020(11),12(5)]),flip(a)].
47504 (x + (((x * y) + (y * x)) * y)) * (x * y) = x * (y * (y * (x * (y * y)))).  [para(37721(a,1),47157(a,1,2,2)),rewrite([37942(9),43804(17),3(15),11(15),402(13),12(12),11446(14),15(11),156(10)]),flip(a)].
47604 x * (x * (y * (x * (y * (x * z))))) = x * (y * (y * z)).  [para(47247(a,1),12(a,1,1)),rewrite([12(3),12(2),12(8),12(7),12(6),12(5)]),flip(a)].
47676 x * (x * (y * (y * (x * (x * z))))) = y * (x * (y * z)).  [para(47365(a,1),12(a,1,1)),rewrite([12(3),12(2),12(8),12(7),12(6),12(5)]),flip(a)].
47700 x * (y * (x * (x * (y * x)))) = y * (x * y).  [para(47365(a,1),11441(a,1,2,2,2)),rewrite([12(12),12(11),12(10),12(13),12(12),12(11),47676(11),47370(9),12(8),12(7),12(6),47676(8),47365(10)])].
47751 x * (y * (y * (x * (y * y)))) = y * (x * x).  [para(37695(a,1),47365(a,2,2)),rewrite([16999(12),12(7),43797(11),27111(8),3(10),9(10),402(8),16020(8),12(2),47501(7),47247(5),47504(9)]),flip(a)].
47784 x * (y * (x * (y * (x * (x * (y * y)))))) = x * (x * (y * (x * x))).  [para(47247(a,1),47365(a,2,2)),rewrite([12(10),12(9),12(8),12(7),9875(8),12(8),12(7),12(6),12(5),47604(9),47247(5),12(11),12(10),12(9),12(8)]),flip(a)].
47804 x * (y * (x * (x * (y * (x * z))))) = y * (x * (y * z)).  [para(47700(a,1),12(a,1,1)),rewrite([12(3),12(2),12(8),12(7),12(6),12(5)]),flip(a)].
47868 x * (y * y) = y * (y * x).  [para(47700(a,1),37695(a,1,2,2,1,2)),rewrite([12(9),12(8),12(7),12(6),47751(9),156(9),12(9),155(10),15(6),3(5),9(5),26672(5),402(3),12(2),12(4),12(3),11441(5),12(7),12(6),12(5),12(4),47751(7)]),flip(a)].
47889 x * (y * (x * (y * (x * x)))) = y * (x * y).  [para(47700(a,1),47157(a,1,2,2)),rewrite([12(7),12(6),12(5),12(4),47604(6),9875(4),47868(12),47700(11),12(10),12(9),12(8),12(7),47804(8)]),flip(a)].
47894 x * (x * (y * (x * x))) = y * x.  [para(47247(a,1),47700(a,2,2)),rewrite([12(9),12(8),12(7),12(6),47889(9),12(9),12(8),12(7),12(6),11439(5),47700(5),11257(4),12(8),12(7),12(6),12(5),47784(8)]),flip(a)].
47896 x * y = y * x.  [para(47700(a,1),47700(a,2,2)),rewrite([12(9),12(8),12(7),12(6),47894(8),12(9),12(8),12(7),12(6),11441(5),12904(6),9875(4),12(8),12(7),12(6),12(5),47604(7),9875(5)])].
47897 $F.  [resolve(47896,a,14,a)].

============================== end of proof ==========================
Douglas S. Stones
  • 21,396
  • 5
    What are you trying to say with this? Are you proving the result by using a computer program? –  Jul 28 '13 at 08:24
  • Next time you present a computerprogram checking the first $10^{20}$ even numbers whether they are the sum of two primes and claim this proofs the Goldbach-conjecture ? (-1) – Peter Sep 19 '23 at 11:28
  • 4
    @Peter At a glance, I don't think that's what this is doing - it seems to be a genuine proof of the result in question. Even though long computerized algebraic proofs are often (as in this case) not human-readable, they are valid proofs (again, unless I'm missing something about what's going on in this one). Personally I think it's interesting to see how long fully-formal proofs wind up being. – Noah Schweber Sep 19 '23 at 17:04
  • 4
    And in this particular case, it highlights (by way of contrast with the top answer) how much of a speed-up is provided by stepping out of purely equational logic into the richer first-order realm. Admittedly I might be in the minority in terms of interest in such things, but I don't think this answer deserves the negative reaction it's gotten. – Noah Schweber Sep 19 '23 at 17:06
  • 5
    @Noah So you upvote answers that "seem to be" proofs, even when not human readable? Answers here are meant to help human readers. It's hard to imagine any human that would be helped by the above 243 line brute-force proof. Generally the purpose of mathematical proofs is not only to compute a Boolean truth value (true or false) but to additionally lend conceptual insight on the matters. Nor is the purpose of the site to contrast human vs machine-generated incomprehensible proofs. If a user posted the above as a human-generated proof it would be heavily downvoted (and likely deleted,). – Bill Dubuque Sep 19 '23 at 21:23
2

Here is an interesting observation:

$2x=(x+x)=(x+x)^4=16x^4=16x$. Thus $14x=0$.

Also $3x=(3x)^4=81x^4=81x$. Thus $78x=0$. Since gcd $(78,14) =2$ we get $2x=0$ for all $x \in R$.

Now one probably needs to look at $x+y=(x+y)^4$ and probably $(x+y+z)=(x+y+z)^4$ or $(x+y+1)=(x+y+1)^4$.

P.S. Actually characteristic 2 is also obtained the following way: $x=x^4=(-x)^4=-x$.

N. S.
  • 134,609
  • 6
    What proved this answer to get eight upvotes? (Well, nine minus one from me.) That $2=0$ in $R$ is almost trivial, but from here to a full proof is a long, long way. –  Jul 28 '13 at 08:36
  • Do you know example of a non trivial ring in which x^4=x for all x.? – Infinity_hunter Jul 29 '22 at 17:43